20090611, 08:32  #342  
"Lucan"
Dec 2006
England
2×3×13×83 Posts 
Quote:
time as Tony. Can you shed any light on the interim residue discrepancy, or would that let the cat out of the bag? 

20090611, 08:58  #343 
Undefined
"The unspeakable one"
Jun 2006
My evil lair
18C3_{16} Posts 
Well it would be easy to test for yourself. Just use a very small number like p=31 or something and print residues every iteration for first and double check modes. You can then compare to a manually computed set of residues. Easy.
Why all the guessing? It can be proved in a matter of minutes! 
20090611, 09:02  #344  
"Nancy"
Aug 2002
Alexandria
2,467 Posts 
Quote:
Alex 

20090611, 09:51  #345  
"Robert Gerbicz"
Oct 2005
Hungary
2773_{8} Posts 
Quote:
Code:
The most number of Mersenne primes in a [x,2.06*x] interval (for the exponent), what we have after the verification for M40M47 are 8 primes. 3 Mersenne primes in 'small' interval: 5 4 Mersenne primes in 'small' interval: 110 5 Mersenne primes in 'small' interval: 536 6 Mersenne primes in 'small' interval: 710 7 Mersenne primes in 'small' interval: 432 8 Mersenne primes in 'small' interval: 162 9 Mersenne primes in 'small' interval: 40 10 Mersenne primes in 'small' interval: 5 I've also counted the number of Mersenne primes in the runs. Note that using the conjecture the expected number of primes is about 48.9 (I've included p=2 in every cases). Code:
26 Mersenne primes: 1 27 Mersenne primes: 0 28 Mersenne primes: 2 29 Mersenne primes: 0 30 Mersenne primes: 1 31 Mersenne primes: 4 32 Mersenne primes: 2 33 Mersenne primes: 3 34 Mersenne primes: 4 35 Mersenne primes: 13 36 Mersenne primes: 15 37 Mersenne primes: 13 38 Mersenne primes: 39 39 Mersenne primes: 37 40 Mersenne primes: 41 41 Mersenne primes: 66 42 Mersenne primes: 72 43 Mersenne primes: 81 44 Mersenne primes: 94 45 Mersenne primes: 100 46 Mersenne primes: 117 47 Mersenne primes: 95 48 Mersenne primes: 134 49 Mersenne primes: 121 50 Mersenne primes: 113 51 Mersenne primes: 134 52 Mersenne primes: 130 53 Mersenne primes: 106 54 Mersenne primes: 93 55 Mersenne primes: 86 56 Mersenne primes: 62 57 Mersenne primes: 56 58 Mersenne primes: 36 59 Mersenne primes: 36 60 Mersenne primes: 27 61 Mersenne primes: 23 62 Mersenne primes: 11 63 Mersenne primes: 10 64 Mersenne primes: 9 65 Mersenne primes: 3 66 Mersenne primes: 5 67 Mersenne primes: 2 68 Mersenne primes: 1 69 Mersenne primes: 1 70 Mersenne primes: 0 71 Mersenne primes: 0 72 Mersenne primes: 0 73 Mersenne primes: 1 Last fiddled with by R. Gerbicz on 20090611 at 09:52 

20090611, 10:45  #346  
"Lucan"
Dec 2006
England
2×3×13×83 Posts 
Quote:
and 2.06 is M47/M40 (exponents of) I don't think we should count both M40 and M47 in that interval. So we should call it a run of 7 rather than 8 primes, much more likely (32%).. Last fiddled with by davieddy on 20090611 at 10:51 

20090611, 11:05  #347  
"Robert Gerbicz"
Oct 2005
Hungary
1,531 Posts 
Quote:
2.06*x>43112609. Last fiddled with by R. Gerbicz on 20090611 at 11:06 

20090611, 11:57  #348  
"Lucan"
Dec 2006
England
2×3×13×83 Posts 
Quote:
This is bias. I still think you are "double counting" like saying the are 8 primes in the interval M1M8, 8 in the interval M8M16...etc. Another way of looking at it is to ask for the probability of finding 8 or more primes AFTER M39 all <= 43112609, giving us anterval of [x,3*x] (roughly) PS I might not have understood exactly what you did in your simulation. Did you generate mock sequences of "M Primes" based on the probability, then take x to be each prime in turn, then count the number of primes >=x and < 2.06*x ? . Last fiddled with by davieddy on 20090611 at 12:33 

20090611, 12:56  #349  
"Robert Gerbicz"
Oct 2005
Hungary
1,531 Posts 
Quote:
Yes, the simulation was that: by probab(p)=if(p%4==3,a=2,a=6);return(1.781*log(a*p)/(p*log(2))) probability I've choosen Mp as prime for every p>2. Note that the interseting fact that probab(3) and probab(5) are higher than 1 so M2,M3,M5 were always in the sequence as primes. The another way would be: simulate also the sequence of primes: if n>1 then by 1/log(n) probability choose n as prime and if it is choosen prime then by probab(p) probability Mp is prime. However this simulation is slower than the previous and I think in theory it gives the same distribution. 

20090611, 13:37  #350  
"Lucan"
Dec 2006
England
2×3×13×83 Posts 
Quote:
in my post which initiated this discussion. Many thanks for your contribution. 

20090611, 14:43  #351 
Aug 2002
Ann Arbor, MI
433 Posts 
Apparently there wasn't supposed to be any interim residue discrepancy (besides the +2 shift between glucas/mlucas and mprime/prime95, which was accounted for). No point in finishing the test if the interim residues show that something already went wrong.

20090611, 14:46  #352 
A Sunny Moo
Aug 2007
USA (GMT5)
6249_{10} Posts 
Are you sure, though, that you weren't simply testing the wrong exponent from the start? If that's the case, then the interim residuals would be different no matter what, and you may as well finish the test rather than letting the already completed work go to waste. No harm in doing a plain old doublecheck on an exponent that needs it.

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